I can see that you're already using an eggbutt. I wouldn't switch that out. I ride all my horses (16 of them) except one in the eggbutt. I do, however, have a real wide mouth one, like this: https://www.bluegrasshorsesupply.com...-snaffle-s.jpg
If you don't want a bit to pinch the corners of your horse's mouth, you can use a bit guard. http://www.nrsworld.com/istarimages/...IT1009!REI.jpg
And the leather strap you asked about that's at the bottom of the bit is called a curb strap. It can serve several different purposes, depending on the bit. On your eggbutt, it can serve any of 3 purposes.
1. Keep the bit from sliding through the horse's mouth
2. Add leverage to the chin to gain collection
3. Help evenly distribute the pressure throughout the entire mouth.
On a shank bit, depending on where it is placed, (either attached to the rings the headstall is attached to, or attached to the rings the reins are attached to) it not only keeps the shanks together, but it also prevents a rider from pulling the bit through the mouth and provides leverage under the chin for collection as well.
Here's a picture similar to the bit I use in the one horse that runs through my eggbutt. Google Image Result for http://www.bitsnmore.com/images/catalog/714.jpg
My gray gelding has a problem when we're working cattle, he will suck in the snaffle bit and slide his tongue over it and pull against it. But my bit is a wide, rubber-wrapped piece of bike chain, so it doesn't pinch in his mouth as bad as this one would.
However, this one may be beneficial to you if your horse is really sensitive in the mouth. Google Image Result for http://www.bitsnmore.com/images/catalog/714.jpg Touch Plus, 5 1/2", sweet iron wrapped, three piece mouth with o-ring center. 6 1/2" cheeks. Designed for the horse with a more sensitive mouth. The slightly longer purchase and cheek add control for rate and supple turns.